Saturday, January 4, 2014

Solving Hilbert’s sixth problem

Invariances of the laws of nature

How is special theory of relativity derived? One starts with the invariance
of the laws of nature to changing inertial reference frames. From this you get either the Lorentz transformation or the Galilean transformation, and use the second postulate, that of the constant value of the speed of light, to select between
the two choices.

But are the laws of nature invariant only to changes in inertial
reference frames? How about a trivial invariance: the laws of nature do not change during time evolution?

And how about another: the
laws of nature do not change if we partition in our mind a larger system into
smaller ones (the composability principle, or the invariance of the laws of
Nature under tensor composition)?

Those statements are completely obvious, but their
mathematical consequences are far from trivial. We need one more ingredient
of a technical nature: a continuity property: if we represent the state of a
physical system with a point in a state/phase/configuration space manifold we
want to be able to compute derivatives, meaning we should be able to define a
tangent plane (after all we will recover the Hamiltonian formalism and the
cotangent bundle).

So now we are ready to begin the journey of deriving the two
products we talked about last time. It will turn out that the symmetric product
describes the ontology and the skew-symmetric one the dynamic. Their compatibility
condition (which will be derived too) will help recover the quantum and
classical mechanics. All this will come out of the two new invariance laws
stated above!!! If you want to follow along in a technical paper (and peek into
the future post contents) use http://arxiv.org/abs/1303.3935
as guidance. Let us begin…

First we consider a set of (abstract) unspecified operations
{o} which include local laws of nature. As concrete examples of such operations
in quantum mechanics, we think of the commutator understood as a product, and the
Jordan product.
At this point we do not assume any properties for those products, or even that
they should exist. How can we mathematically state that the set {o} remains
invariant under time evolution? The idea is that of an isomorphism: “o“ at time
t must be isomorphic with “o” at time t+delta t because the isomorphism preserves
the algebraic relationships. If T is a time translation operator (T A(t) = A(t+
delta t)) the isomorphism can be written as:

T [G o H] = [T(G) o T(H)]

Or equivalently:

(G o H)(t +delta t) =
G (t +delta t) o H(t+delta t)

This is the only consequence we can derive from the
invariance of the laws of nature under time evolution.

Now consider infinitesimal time evolutions and use
the ability to derivate. This introduces a
tangent plane at “t” and a (particular) vector field associated with (a
particular) time evolution.

If T_epsilon is a particular infinitesimal (time) translation
operator we have:

T_epsilon [G(t) o H(t)] = [T_epsilon G(t) o T_epsilon H(t)]

If rho is one of the products in the set {o} corresponding
to the time translation transformation T_epsilon, we can construct the
following product between a distinguished f and any g:

(Here T_epsilon and “f” are not arbitrary, but depend on each
other. “f” in general corresponds to a particular Hamiltonian, and ρ corresponds to the Poisson bracket in
classical mechanics and the commutator in quantum mechanics.)

We generalize the product ρ for all f’s and g’s by repeating the argument for all
conceivable dynamics. To make sure the domains of f and g are identical and
well behaved, in case of pathologies, we can restrict the domain of g’s to the
span of all possible f’s.

From the invariance of the laws of Nature under time evolution
we have:

The Leibnitz identities turn out to be of critical
importance as we will see in subsequent posts.

In summary, the fact that the laws of nature do not change during time evolution along
with differentiation demands the existence of left and right Leibnitz identities for an abstract product ρ.

As an important note, we do not know at this time that ρ is necessarily skew-symmetric. If we
assume this from the very beginning the whole derivation is much shorter.

Next time we’ll show that dynamic alone is not enough and we
will need a supporting product which in the end will show that it corresponds to
observables (ontology). Then we’ll establish the most general relations between
dynamic and ontology.

Thanks Lawrence, I had an enjoyable holiday, hope you had the same. I have new results in different stages of writing. Now I am doing this current series on results I got more than a year ago. I am reading about field theory in curved space-time and there the best approach is the same algebraic one I am presenting. The major issue is that of the lack of a distinguished vacuum state and a distinguished Hilbert space representation. Looking forward to your material.